Theorem ontric3g 42575

Description: For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)

Assertion

Ref	Expression
ontric3g	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ontric3g

Step	Hyp	Ref	Expression
1		orcom 866	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥))
2	1	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
3		onsseleq 6404	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
4		ontri1 6397	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
5	2, 3, 4	3bitr2d 306	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ ¬ 𝑥 ∈ 𝑦))
6	5	con2bid 353	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
7	6	ancoms 457	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
8	4	ancoms 457	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
9		ontri1 6397	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
10	8, 9	anbi12d 629	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥)))
11		eqss 3996	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦))
12		ioran 980	. . . 4 ⊢ (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥))
13	10, 11, 12	3bitr4g 313	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
14		equcom 2019	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
15	14	orbi2i 909	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦))
16	15	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
17		onsseleq 6404	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
18	16, 17, 9	3bitr2d 306	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ ¬ 𝑦 ∈ 𝑥))
19	18	con2bid 353	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))
20	7, 13, 19	3jca 1126	. 2 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥))))
21	20	rgen2 3195	1 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   ∈ wcel 2104  ∀wral 3059   ⊆ wss 3947  Oncon0 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367

This theorem is referenced by: (None)